Relative Nakayama-Zariski decomposition and minimal models of generalized pairs

TL;DR

This paper develops properties of the relative Nakayama-Zariski decomposition and applies them to establish the existence of minimal models or Mori fiber spaces for lc generalized pairs, extending key results in the minimal model program.

Contribution

It introduces new properties of the relative Nakayama-Zariski decomposition and proves the existence of minimal models or Mori fiber spaces for lc generalized pairs, extending previous results.

Findings

Existence of log minimal models or Mori fiber spaces for lc generalized pairs polarized by an ample divisor.

Extension of Hashizume-Hu's results to generalized pairs.

Establishes that certain lc generalized pairs have either a log minimal model or a Mori fiber space.

Abstract

We prove some basic properties of the relative Nakayama-Zariski decomposition. We apply them to the study of lc generalized pairs. We prove the existence of log minimal models or Mori fiber spaces for (relative) lc generalized pairs polarized by an ample divisor. This extends a result of Hashizume-Hu to generalized pairs. We also show that, for any lc generalized pair $(X,B+A,{\bf{M}})/Z$ such that $K_X+B+A+{\bf{M}}_X\sim_{\mathbb R,Z}0$ and $B\geq 0,A\geq 0$, $(X,B,{\bf{M}})/Z$ has either a log minimal model or a Mori fiber space. This is an analogue of a result of Birkar/Hacon-Xu and Hashizume in the category of generalized pairs, and is later shown to be crucial to the proof of the existence of lc generalized flips in full generality.